Quantification of non-Markovian effects in the Fenna-Matthews-Olson complex

The excitation energy transfer dynamics in the Fenna-Matthews-Olson complex is quantified in terms of a non-Markovianity measure based on the time evolution of the trace distance of two quantum states. We use a system description derived from experiments and different environmental fluctuation spectral functions, which are obtained either from experimental data or from molecular dynamics simulations. These exhibit, in all cases, a nontrivial structure with several peaks attributed to vibrational modes of the pigment-protein complex. Such a structured environmental spectrum can, in principle, give rise to strong non-Markovian effects. We present numerically exact real-time path-integral calculations for the transfer dynamics and find, in all cases, a monotonic decrease of the trace distance with increasing time which renders a Markovian description valid.

Figure 1

Spectral density function of Adolphs and Renger [Eq. (4)] for different Lorentzian peak widths $\gamma$ centered at ${\omega }_H=180$ cm${}^{-1}$.

Figure 2

Structural arrangement of the seven BChl molecules (black numbers) in FMO (Chl. tepidum) [40, 52] superposed with a schematic representation of the delocalization patterns of the different excitons (colored shading, italic red numbers). The exciton numeration is in ascending energy order. The two main excitation transfer routes are indicated by the green and black thin arrows. Entrance and exit sites are indicated by blue thick arrows.

Figure 3

Spectral density function of Adolphs and Renger [Eq. (4)] with no localized vibrational mode (solid black line) and for ${\omega }_H$ in resonance with excitonic energy differences: ${\omega }_H=190.8$ cm${}^{-1}$ (dashed red line) and ${\omega }_H=211.0$ cm${}^{-1}$ (dash-dotted blue line). $\gamma =1$ cm${}^{-1}$ in both cases.

Figure 4

Spectral density function of Kreisbeck and Kramer for the FMO complex in the form of a sum of shifted Drude-Lorentz peaks [Eq. (7)] with the parameters listed in Table 2.

Figure 5

Site-dependent spectral density functions $G_j\left(\omega \right)$ of Aghtar et al. for the FMO complex as determined from molecular dynamics simulations [25] with water as a solvent at 300 K (solid black line) and with a glycerol:water (65:35) mixture as a solvent at 310 K (dashed blue line).

Figure 6

Time evolution of the trace distance [Eq. (9)] for the spectral density function derived by Adolphs and Renger [Eq. (4)] in the absence of any localized vibrational mode ($\gamma =0$).

Figure 7

Time evolution of the trace distance [Eq. (9)] for the spectral density function derived by Adolphs and Renger [Eq. (4)] as a function of the width $\gamma$ of the Lorentzian peak centered at (a) 180, (b) $190.8$, and (c) $211.0$ cm${}^{-1}$. Left and right columns correspond to temperatures of 77 and 300 K, respectively.

Figure 8

Time evolution of the trace distance [Eq. (9)] for the spectral density derived by Kreisbeck and Kramer [Eq. (7)] with $n=3$ (thin black lines) and $n=11$ (thick red lines) at 77 K (solid lines) and 300 K (dashed lines).

Figure 9

Time evolution of the trace distance [Eq. (9)] for the spectral density derived by Aghtar et al. [Eq. (8)] with water as a solvent at 300 K (solid black line) and with a glycerol:water (65:35) mixture as a solvent at 310 K (dashed blue line).